After the death of Henry Purcell in 1695, it was almost 300 years before Britain produced another composer (Edward Elgar) of international standing. This subsequently led one German commentator to describe Britain as ‘Das Land ohne Musik’. In a similar vein, the period 1750–1830 is described in this book as being one of British ‘mathematical stagnation’, and its 466 pages describe 19^{th} century developments that put Britain on the road to recovery from being ‘Das Land onhe Mathematik’.

Various reasons are given for the weak state of British mathematics in the period 1750–1830. First was the almost patriotic adherence to Newtonian calculus (based upon fluxions and fluents). But there was also prolonged dependence on the geometrically based methods of Euclid. Another factor was the low standing of mathematics in the university curricula and the lack of structured support for mathematical research. Moreover, until 1865, Britain had no major mathematical society nor any serious journal for the dissemination of mathematical research. Non-mathematical reasons (given in this book) include Britain’s peripheral geographical position, and the long war with France that intensified its cultural isolation from mainland Europe

*Mathematics in Victorian Britain* is the product of sixteen authors, and it consists of eighteen chapters that can be read in any order. It presents researches on the history of mathematics in Victorian Britain ‘that would otherwise be out of reach to the general reader’. As such, it claims to be ‘the first general survey of the mathematics of the Victorian period’ that is accessible to the mathematically inclined general reader.

The first six chapters follow the geographical theme of mathematical life in London, Cambridge, Oxford, Scotland, Ireland and parts of the British Empire. They chart the growth of institutional development of mathematics as a profession during the 19^{th} century, and they describe changes in the teaching of mathematics in the principal British centres of mathematical education. These chapters therefore provide a context for the discussion of mathematics and mathematicians featured in ten of the remaining chapters. These deal with the themes of Victorian astronomy, calculating engines, statistics, calculus and differential equations, geometry, algebra, logic, combinatorics and an overview of applied mathematics.

Some of these themes are presented with emphasis on social context (e.g. statistics and the chapter on Victorian mathematical journals), while others concentrate more deeply on the relevant mathematics (e.g. combinatorics, calculus and differential equations in Victorian Britain). There is a high standard of expository writing throughout the book, which is made all the more attractive by an abundance of illustrations from the Victorian period.

The book opens with the declaration that the reign of Queen Victoria (1837-1901) witnessed a ‘dramatic renaissance’ in British mathematics. In support of this claim, there is a recitation of notable names from the period, including those of Babbage, Maxwell, Cayley, Sylvester, Russell, Hamilton, Boole, De Morgan and various others. Innovative ideas arising from such mathematicians are said to include vectors, quaternions, matrices, Boolean algebra and aspects of applied mathematics and statistics. In moderation, however, Victorian mathematics is more soberly described as ‘a mixture of the antiquated and the progressive’.

Speaking of ‘moderation’ brings to mind the very last chapter, written by Jeremy Gray, who outlines several fundamental principles of historiography. In this context, compelling arguments are provided to the effect that ‘British pure mathematics of the 19^{th} century has been overrated to the detriment of historical writing on the subject’. In the process, Gray exposes weaknesses in the work of two of the most notable British pure mathematicians of the Victorian period (Cayley and Sylvester), of whom he says:

If the names of Cayley and Sylvester have cropped up so often in this chapter, it is in fair part because there are not many other people to talk about.

Of the period covered by the 16 chapters of this book, Gray also says:

I suggest that we should not look at the story of pure mathematics in Britain in the 19^{th} century as a success story, but as a particular kind of failure — or, if you prefer, of partial success against great odds.

However, whatever the weaknesses of British mathematics in the Victorian period, the abundance of information on the social or institutional circumstances from which it arose, and the information about the major and minor figures who created it, make this book a most enjoyable read.

Peter Ruane taught mathematics to people of greatly varying abilities between the ages of 5 (kindergarten) and 55 (mature undergraduates) — or, rather, he tried to teach them how to learn mathematics.